process

202 J. CuntzWe denote by N the set of natural numbers excluding 0. N will normally beregarded as a semigroup with multiplication.The natural analogue, for a semigroup S, of an unitary representation of a group isa representation of S by isometries, i.e. by operators s g , g 2 S on a Hilbert space thatsatisfy sg s g D 1.On the Hilbert space `2.N/ consider the isometries s n , n 2 N and v k , k 2 Ndefined byv k . m / D mCk ; s n . m / D mnwhere m ;m2 N denotes the standard orthonormal basis.We have s n s m D s nm and v n v m D v nCm , i.e. s and v define representations of N and N respectively, by isometries. Moreover we have the following relations n v k D v nk s nwhich expresses the compatibility between multiplication and addition.In other words, the s n and v k define a representation of the ax C b-semigroup 1 kP N D j n 2 N ;k2 N0 nover N, where the matrix0n 1k is represented by .v k s n / .Note that, for each n, the operators s n ;vs n ;:::;v n 1 s n generate a C -algebraisomorphic to O n , [4].3 A purely infinite simple C -algebra associated with theax C b-semigroupThe C -algebra A generated by the elements s n and v considered in Section 2 containsthe algebra K of compact operators on `2.N/. Denote by u, resp. u k the image of v,resp. v k in the quotient A=K. We also still denote by s n the image of s n in the quotient.Then the u k are unitary and are defined also for k 2 Z and they furthermore satisfythe characteristic relation P n 1kD0 uk e n u k D 1 where e n D s n sn denotes the rangeprojection of s n . This relation expresses the fact that N is the union of the sets ofnumbers which are congruent to k mod n for k D 0;:::;n 1.We now consider the universal C -algebra generated by elements satisfying theserelations.Definition 3.1. We define the C -algebra Q N as the universal C -algebra generated byisometries s n , n 2 N with range projections e n D s n sn , and by a unitary u satisfyingthe relationsXn 1s n s m D s nm ; s n u D u n s n ; u k e n u k D 1for n; m 2 N .kD0